Is it possible to symmetric weight-paint an asymmetric mesh?
EDIT: According to person asking this question, this is the exact opposite of what they wanted, all i can say is just tick it on instead of off, but they said that doesn't work, i shall keep this up for those who find it though.

This is because X Mirroring is on, to turn it off go to into the Weight Painting mode on your desired object. 

On the left Toolshelf T select the Options(1) tab.

Under Options(2) Uncheck the checkbox labeled as "X Mirror"(3)



Your weight painting should be fine now! 

Tip: open up a new file, without changing anything (other than what i'm saying now), even moving the camera, repeat the steps above and then with the mouse over 3D view, press CTRL+U and then press "Save Startup File"

This makes your blender startup file have what you just did already done, so you don't have to do it again.